Heterogeneity of Investors and Asset Pricing in a Risk-Value World

The investor minimizes his risk subject to the budget contraint E(Rk) = 1
and the return constraint E(R) > R*. The solution gives the efficient return
sharing rule.

The ratio A/Wq is crucial for risk measurement if 7 > -∞. Depending
on the investor, the ratio A/Wq might be independent of Wq or not. Define

A* _ A R

W = W ^- 1 - 7 ’

Then

ʌ 1 - 7 /A*    Rλ⁷

^e[F(^r)] = ^--₇⅛(- ₊ -) .         (19)

Investors differ in terms of Wq /A* for 7 > -∞ resp. BWq for 7 = -∞
and the required expected return R*. Hence the sharing constants of two
investors differ because of differences in these parameters. Applying Lemma
1 shows that the sharing constant of an investor declines when Wq/A* resp.
BWq or the required expected return R* increases.

This proves

Proposition 5 ; Given the pricing kernel, the difference between the shar-
ing constants of investors i and j, (s_i - s₃), is monotonically increasing in
(W_0j∕A* - W_o√A*) for 7 > -∞ resp. (B_jW(p- - BW) for 7 = -∞ and
(R - R)■

Before we discuss the sharing rules of these investors, it is helpful to
understand the impact of Wq/A* resp. BWq. The curvature of the risk
function, -f(R)/f (R), similar to absolute risk aversion, increases monoton-
ically in Wq/A* resp. BWq. Therefore we denote Wq/A* resp. BWq as the
investor’s risk sensitivity. With 1 > 7 > -∞, risk sensitivity is higher for

24

<<   21   22   23   24   25   26   27   >>

More intriguing information